438 ON CONVECTION CURRENTS IN A [412
of a former investigation where the fluid was supposed from the beginning to be incompressible but of variable density *.
Returning to the consideration of a finite conductivity, we have again to distinguish the cases where JB is positive and negative. When /9 is negative (higher temperature below) both values of n in (17) are real and one is positive. The equilibrium is unstable for all values of I* + w2 and of s. If y3 be positive, n may be real or complex. In either case the real part of n is negative, so that the equilibrium is stable whatever Z2 + m2 and s may be.
When (3 is negative (— /3'), it is important to inquire for what values of I2 + in* the instability is greatest, for these are the modes which more and more assert themselves as time elapses, even though initially they may be quite subordinate. That the positive value of n must have a maximum appears when we observe it tends to vanish both when I2 + in2 is small and also when Z2 + ma is large. Setting for shortness I* + m2 + s~ = a, we may write (17)
n2o- + me** - fly (ff - s2) = 0, ..................... (20)
and the question is to find the value of a for which n is greatest, s being supposed given. Making dn/d<r = 0, we get on differentiation
n2 + 2n.*o--/3V = 0; ........................... (21)
and on elimination of ?i2 between (20), (21)
(22)
Using this value of n in (21), we find as the equation for a
^-l-££ ............................... (23)
a /c2<r4 v
When K is relatively great, cr = 2s2, or
£2 + w2 = s2 .................................. (24)
A second approximation gives
The corresponding value of n is
" ............................ (26)
The modes of greatest instability are those for which s is smallest, that is equal to 7r/£, and
77" o 'V
J. + „* = _.+__ ......................... (27)
* Proc. Lond. Math. Soc. Yol. xiv. p. 170 (1883) ; Scientific Papers, Vol. n. p. 200.onductivity falls within the scope dv 1 diff dw 1 d™ed force,